Question

How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where $\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that:

(1) $C_{i+1} = \{i\}$ and $C_i$ is closed and cofinal in $i$ if $i$ is a limit ordinal.

(2) If $i$ is a limit point of $C_j$, then $C_i = C_j \cap i$.

(3) There is no club $C$ (a subset of $\kappa$) such that for all limit points $i$ in $C$ the equality $C_i= C \cap i$ holds.

Answer 1

In general, one cannot force the failure of $\square(\kappa)$ at a fixed cardinal $\kappa$. Indeed, if $\kappa$ is any regular uncountable cardinal which is not weakly compact in $L$, then there is a nontrivial $\square(\kappa)$ sequence which is moreover constructible. The fact that $\kappa$ is not weakly compact in $L$ cannot be destroyed by forcing. On the other hand, $\square(\kappa)$ always fails at a weakly compact cardinal.